Download 4 - edl.io

4.10 Write Quadratic Functions and Models
Goal  Write quadratic functions and models.
Your Notes
VOCABULARY
Best-fitting quadratic model
The model given by performing quadratic regression on a calculator
Example 1
Write a quadratic function in vertex form
Write a quadratic equation for the parabola shown.
y  a(x  h)2  k
Use the vertex form
because the vertex
is given.
y  a(x  2_)2 _ 3_
Substitute.
Use the other given point, (_0_ , _5_), to find a.
_5_  a(_0  2_)2 _ 3_
_2  a _
Substitute for x and y.
Solve for a.
A quadratic function for the parabola is
_y  2(x  2)2  3_ .
Example 2
Write a quadratic function in intercept form
Write a quadratic equation for the parabola shown.
y  a(x  p)(x  q)
Use the intercept
form because the
x-intercepts are
given.
y  a(x  3_)(x  2_)
Substitute.
Use the other given point, (_2_ , _4_), to find a.
_4_  a(_2 + 3_)(2  2_)
_1_ = a
Substitute for x and y.
Solve for a.
A quadratic function for the parabola is
_y = (x  3)(x 2) .
Your Notes
Checkpoint Write a quadratic function whose graph has the given
characteristics.
1.
x-intercepts: 2, 1 point on graph: (1, 4)
y  2(x  2)(x  1)
2.
vertex: (2,1) point on graph: (0, 4)
y  3 (x  2)2  1
4
Example 3
Write a quadratic function in standard form
Write a quadratic function in standard form for the parabola that passes through
the points (2, 6), (0, 6) and (2, 2).
Substitute the coordinates of each point into y  ax2  bx  c to obtain a system of three
linear equations.
_6_  a(_2_)2  b(_2_)  c Substitute for x and y.
Equation 1
_6_  _4a  2b  c
2
Substitute for x and y.
_6_  a(_0_)  b(_0_)  c
Equation 2
_6_  _c_
2
Substitute for x and y.
_2_  a(_2_)  b(_2_)  c
Equation 3
_2_  _4a  2b  c_
Rewrite the system as a system of two equations.
Substitute for c.
_4a  2b  6_  _6_
Substitute 6 for c
in equation 1.
_4a  2b  _12_
_4a  2b  6_  _2_
Revised Equation 1
Substitute for c.
Substitute 6 for c
in equation 3.
Revised Equation 3
_4a  2b  _4_
Solve the system consisting of revised equations 1 and 3.
Revised Equation 1
_4a  2b  12_
Revised Equation 3
_4a  2b  4_
Add Equations.
_8a
_ 16_
Solve for a.
a  _2_
So _4( 2)  2b  _4_, which means b  _2_ .
A quadratic function for the parabola is
_y  2x2  2x  6_.
Your Notes
Example 4
Solve a multi-step problem
Baseball The table shows the height of a baseball hit, with x representing the time
(In seconds) and y representing the baseball’s height (In feet). Use a graphing calculator
to find the best-fitting model for the data.
Time, x
0
2
4
6
8
Height, y
3
28
40
37
26
Solution
Enter the data into two lists
of a graphing calculator.
Use the quadratic regression
model feature to find the
best-fitting quadratic model
for the data.
Make a scatter plot of
the data.
Check how well the
model fits the data by
graphing the model and
the data in the same
viewing window.
QuadReg
y  ax2  bx  c
a  _1.553571429_
b  _15.17857143_
c  _3.371428571_
The best fitting quadratic model is
_y  1.55x2  15.2x  3_.
Your Notes
Checkpoint Complete the exercises below.
3.
Write a quadratic function in standard form for the parabola that passes through
(1, 5), (2, 1) and (3, 1).
y  x2  3x  1
4. Use a graphing calculator to find the best-fitting model for the data in the table.
Time, x
0
2
4
6
8
Height, y
4
23
30
25
7
y  1.54x2  12.7x  4.
Homework
________________________________________________________________________
________________________________________________________________________